OneTaskBy Tim King · Updated
Regular Polygon Calculator
Hexagon and Octagon Measurements, Interior Angle, Apothem and Diagonals
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How the Regular Polygon Calculator works
This regular polygon calculator works out every measurement of a regular polygon, one with all sides and angles equal, from the number of sides and any one value: side length, perimeter, area, circumradius or apothem. It covers hexagons, octagons and anything up to 1,000 sides, and shows the working.
A regular polygon can be split into n identical isosceles triangles meeting at the centre. Each has a central angle of 360° ÷ n, a base equal to the side length and a height equal to the apothem, the distance from the centre to the middle of a side. The circumradius is the distance from the centre to a corner. Everything else follows from those triangles.
How the regular polygon calculator works
The core formula is area = n s² ÷ (4 tan(180° ÷ n)). The circumradius is s ÷ (2 sin(180° ÷ n)) and the apothem is s ÷ (2 tan(180° ÷ n)). Each interior angle is (n − 2) × 180° ÷ n and the polygon has n(n − 3) ÷ 2 diagonals. If you give a value other than the side, the calculator rearranges the matching formula to find s first.
First example. A hexagon with 4 cm sides has an area of 41.57 cm², an apothem of 3.46 cm and a circumradius of exactly 4 cm, because a hexagon is six equilateral triangles.
Second example. A stop-sign octagon 75 cm across the flats has an apothem of 37.5 cm, so each side is 2 × 37.5 × tan(22.5°) ≈ 31.07 cm.
How many diagonals does a decagon have? 10 × 7 ÷ 2 = 35.
Angles, diagonals and area with side length 1 for polygons with 3 to 26 sides
| Sides | Name | Interior angle | Angle sum | Diagonals | Area (side 1) |
|---|---|---|---|---|---|
| 3 | triangle | 60° | 180° | 0 | 0.4330 |
| 4 | square | 90° | 360° | 2 | 1.0000 |
| 5 | pentagon | 108° | 540° | 5 | 1.7205 |
| 6 | hexagon | 120° | 720° | 9 | 2.5981 |
| 7 | heptagon | 128.57° | 900° | 14 | 3.6339 |
| 8 | octagon | 135° | 1080° | 20 | 4.8284 |
| 9 | nonagon | 140° | 1260° | 27 | 6.1818 |
| 10 | decagon | 144° | 1440° | 35 | 7.6942 |
| 11 | hendecagon | 147.27° | 1620° | 44 | 9.3656 |
| 12 | dodecagon | 150° | 1800° | 54 | 11.1962 |
| 13 | 13-gon | 152.31° | 1980° | 65 | 13.1858 |
| 14 | 14-gon | 154.29° | 2160° | 77 | 15.3345 |
| 15 | pentadecagon | 156° | 2340° | 90 | 17.6424 |
| 16 | hexadecagon | 157.5° | 2520° | 104 | 20.1094 |
| 17 | 17-gon | 158.82° | 2700° | 119 | 22.7355 |
| 18 | 18-gon | 160° | 2880° | 135 | 25.5208 |
| 19 | 19-gon | 161.05° | 3060° | 152 | 28.4652 |
| 20 | icosagon | 162° | 3240° | 170 | 31.5688 |
| 21 | 21-gon | 162.86° | 3420° | 189 | 34.8315 |
| 22 | 22-gon | 163.64° | 3600° | 209 | 38.2533 |
| 23 | 23-gon | 164.35° | 3780° | 230 | 41.8344 |
| 24 | 24-gon | 165° | 3960° | 252 | 45.5745 |
| 25 | 25-gon | 165.6° | 4140° | 275 | 49.4738 |
| 26 | 26-gon | 166.15° | 4320° | 299 | 53.5323 |
Tips
- Across the flats or across the corners? Across the flats is twice the apothem; across the corners is twice the circumradius.
- Exterior angles always sum to 360°. That makes 360 ÷ n the quickest angle check.
- Large n approaches a circle. At 1,000 sides the area is within 0.001% of the circle's.
FAQ
How do you find the interior angle of a regular polygon?
Subtract 2 from the number of sides, multiply by 180 and divide by the number of sides. A pentagon has (5 − 2) × 180 ÷ 5 = 108° at every corner. The exterior angle is 180° minus that, or simply 360° divided by the number of sides.
What is an apothem?
The apothem is the line from the centre of a regular polygon to the midpoint of one side, meeting it at a right angle. It is the radius of the largest circle that fits inside. Area is half the perimeter times the apothem, a quick check on any answer this calculator gives.
How do you find the side length of a hexagon from its area?
Rearrange the area formula: s = √(2A ÷ (3√3)). A hexagon of 100 cm² has sides of about 6.2 cm. Set the number of sides to 6, choose Area in the second box, and the calculator does the rearranging for any polygon, not just hexagons.
Why are hexagons used for tiles and honeycombs?
Only triangles, squares and hexagons tile a flat surface with identical regular shapes, and of those the hexagon encloses the most area for the least perimeter. Bees use less wax for the same storage, and hexagonal tiles need less grout line per square metre than squares.
See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.
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